Exact stability analysis of LTI systems with multiple distributed delays based on the Dixon resultant

This paper investigates the stability problem of the linear time-invariant (LTI) system with multiple distributed delays from a new perspective. First, an equivalent system with four discrete delays is proposed from the stability perspective. A novel frequency sweeping methodology is deployed to determine the kernel and offspring hypersurfaces (KOH) of the system. For this, the Dixon resultant theory is employed to determine the upper and lower frequency bounds of the system's imaginary spectra. Secondly, the standing root case is particularly examined to detect the precise stationary root boundary (SRB), which constitutes the non-conservative and exhaustive potential stability changing curves (PSCC) together with the KOH. The 2-D cross-section of the stability map is established by applying the cluster treatment of characteristic roots (CTCR) methodology. We show the effectiveness of our proposed procedures over one example case study. Finally, the whole stability chart is confirmed with the numerical method named Quasi-Polynomial mapping-based Root-finder (QPmR) and the Simulink-based simulation.